Finite Group Databasebeta

← #20273701

Subgroups — #20273701

Order 6784.

280 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810934C212yesmaximal, complemented
2#16810548C211yesmaximal, characteristic, complemented
3#16810357C212yesmaximal, complemented
4#16810347C211yesmaximal, characteristic, complemented
5C53×#534C211yesmaximal, characteristic
6C2×C4⋊Dic53C2211yescharacteristic, complemented
7C2×C22⋊Dic53—21no—
8#13133150—22no—
9#13133138—22no—
10C22⋊#88322C2212yescomplemented
11#13133124—21no—
12#13133124C2211yescharacteristic, complemented
13C22×D4×C53—21no—
14#123×C53—22no—
15#121×C53—21no—
16#121×C53C2212yes—
17#120×C53—22no—
18C2×C22⋊C4×C53C2211yescharacteristic
19C24×C53—21no—
20C24×C53D411yescharacteristic
21C2×D4×C53—22no—
22C2×D4×C53—22no—
23C2×D4×C53—21no—
24C2×D4×C53—42no—
25C2×D4×C53—42no—
26C2×D4×C53D412yes—
27C22×C4×C53—21no—
28C22×C4×C53D411yescharacteristic
29C22×C4×C53C2311yescharacteristic
30C22×Dic53—21no—
31C22×Dic53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53D412yes—
34C22⋊C4×C53—22no—
35C22⋊C4×C53—22no—
36C22⋊C4×C53—22no—
37C22⋊Dic53—42no—
38C4⋊Dic53—22no—
39C2×#88322—22no—
40C23×C53—21no—
41C23×C53—21no—
42C23×C53—22no—
43C23×C53—41no—
44C23×C53—41no—
45C23×C53—42no—
46C23×C53—81no—
47C23×C53—81no—
48C23×C53D812yes—
49C23×C53C2×D411yescharacteristic
50D4×C53—42no—
51D4×C53—44no—
52D4×C53—42no—
53D4×C53—42no—
54D4×C53—81no—
55C2×C4×C53—22no—
56C2×C4×C53—21no—
57C2×C4×C53—41no—
58C2×C4×C53—42no—
59C2×C4×C53—41no—
60C2×C4×C53C2×D412yes—
61C2×Dic53—41no—
62C2×Dic53—41no—
63C2×Dic53—41no—
64C2×Dic53—41no—
65C2×Dic53—41no—
66C2×Dic53—41no—
67#88322—42no—
68C22×C53—21no—
69C22×C53—21no—
70C22×C53—22no—
71C22×C53—42no—
72C22×C53—41no—
73C22×C53—41no—
74C22×C53—42no—
75C22×C53—42no—
76C22×C53—42no—
77C22×C53—81no—
78C22×C53—84no—
79C22×C53#12011yescharacteristic
80C22×C53C2×D811yescharacteristic
81C22×C53AGL(1,ℤ/8)11yescharacteristic
82C4×C53—22no—
83C4×C53—42no—
84C4×C53—41no—
85Dic53—81no—
86Dic53—81no—
87#1607—531nomaximal, Sylow
88C2×C53—21no—
89C2×C53—21no—
90C2×C53—42no—
91C2×C53—82no—
92C2×C53#44711yescharacteristic
93C2×C53#45511yescharacteristic
94C2×C53#45811yescharacteristic
95#534D5311yescharacteristic
96#480—532no—
97#397—531no—
98#327—532no—
99#323—531no—
100C53#160711yescharacteristic, complemented, Sylow
101C22×D4—21no—
102#123—22no—
103#121—21no—
104#121C2×D5312yes—
105#120—22no—
106C2×C4⋊C4—531no—
107C2×C22⋊C4—1061no—
108C2×C22⋊C4C2×D5311yescharacteristic
109#107—1062no—
110#102—1062no—
111C22⋊C8—532no—
112#95—531no—
113#95—1061no—
114C24—21no—
115C24C22⋊D5311yescharacteristic
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—21no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4C22⋊D5312yes—
122C22×C4—21no—
123C22×C4—1061no—
124C22×C4—1061no—
125C22×C4C22⋊D5311yescharacteristic
126C22×C4C22×D5311yescharacteristic
127C2×C8—1062no—
128C4⋊C4—21no—
129C4⋊C4—1062no—
130C4⋊C4C22⋊D5312yes—
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—22no—
134C22⋊C4—2122no—
135C23—21no—
136C23—21no—
137C23—22no—
138C23—41no—
139C23—41no—
140C23—42no—
141C23—81no—
142C23—81no—
143C23#1171992212yes—
144C23C2×C22⋊D5311yescharacteristic
145D4—42no—
146D4—44no—
147D4—42no—
148D4—42no—
149D4—81no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—2121no—
156C2×C4—2121no—
157C2×C4—2121no—
158C2×C4—2121no—
159C2×C4—2121no—
160C2×C4—2121no—
161C2×C4C2×C22⋊D5312yes—
162C8—2122no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—81no—
173C22—84no—
174C22#1313318611yescharacteristic
175C22C2×#1171992211yescharacteristic
176C22#1313324311yescharacteristic
177C4—22no—
178C4—42no—
179C4—41no—
180C4—4241no—
181C4—4241no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—82no—
186C2#1681075711yescharacteristic
187C2#1681080111yescharacteristic
188C2#1681081311yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#534C2
socleC22×C53#120
hypercenter#534D53
hypocenterC53#1607
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #16810934C2semidirect
2. #16810548C2semidirect
3. #16810357C2semidirect
4. #16810347C2semidirect
6. C2×C4⋊Dic53C22semidirect
10. C22⋊#88322C22semidirect
12. #13133124C22semidirect
100. C53#1607semidirect